Imagine an extreme case where your board is so convex that the bottom is actually half a cylinder. You won't be going far with that one.

The extreme of concave is harder to imagine, yet it is easy to see that when you have an exaggerated concave, you will have the heelside of the board vertical (which will grip in the water like a keel) and the toeside horizontal (which will deflect the water).
The concave would then help you go upwind better. It probably has other effects that are harder to picture.